In this lecture, we prepare the way for the notion of local logic by studying the ways that classifications give rise to “regular theories.” These theories can be seen as an idealized version of the scientific laws supported by a given closed system. The adjective “regular” refers to the purely structural properties that any such theory must satisfy. Any theory with these properties can be obtained from a suitable classification. At the end of the lecture, we will return to the question of how different scientific theories, based on different models of the phenomena under study, can be seen as part of a common theory. We will see conditions under which this obtains.

Theories

One way to think about information flow in a distributed system is in terms of a “theory” of the system, that is, a set of known laws that describe the system. Usually, these laws are expressed in terms of a set of equations or sentences of some scientific language. In our framework, these expressions are modeled as the types of some classification. However, we will not model a theory by means of a set of types. Because we are not assuming that our types are closed under the Boolean operations, as they are not in many examples, we get a more adequate notion of theory by following Gentzen and using the notion of a sequent.